Problem 1 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$2 x y\left(5 x y^{2}+3 x^{2} y^{3}\right)$$

Step-by-Step Solution

Answer

The product is \(10x^2y^3 + 6x^3y^4\).

Step 1: Identify the Binomials

We are given the expression to multiply: \(2xy(5xy^2 + 3x^2y^3)\). The terms inside the parentheses form a binomial.

Step 2: Apply the Distributive Property

Use the distributive property which states \( a(b + c) = ab + ac \) to expand the expression. Distribute \(2xy\) to each term in the binomial inside the parentheses.

Step 3: Multiply Each Term

First, multiply \(2xy\) by \(5xy^2\): \[2xy imes 5xy^2 = 10x^1y^1 imes x^1y^2 = 10x^{2}y^{3}\] Next, multiply \(2xy\) by \(3x^2y^3\):\[2xy imes 3x^2y^3 = 6x^1y^1 imes x^2y^3 = 6x^{3}y^{4}\]

Step 4: Combine the Terms

Write the expression as the sum of the expanded terms, resulting in: \(10x^2y^3 + 6x^3y^4\).

Key Concepts

Distributive Property

Imagine you're unpacking a suitcase. Just as you take each item out, one at a time, when multiplying binomials, we use the distributive property to "distribute" or unpack the multiplier across each part inside the parentheses. When given something like \(a(b + c)\), you're essentially dealing with \(a \cdot b + a \cdot c\).
This approach ensures that each component of the binomial interacts with the factor outside, ensuring no term is left behind. It's all about careful placement and combination, much like setting each of your suitcase items in the right place.

• The distributive property is fundamental to multiply binomials effectively.
• Always distribute the term outside the parentheses to every term inside.
• It's a consistent method used in a variety of algebraic problems.

Polynomial Expansion

Polynomial expansion is akin to spreading out, or expanding, a compressed idea into a fuller, detailed expression. Starting with something compact like \((5xy^2 + 3x^2y^3)\) and expanding it using another term, such as \(2xy\), you build upon its complexity by allowing every term to interact with each other.
In our problem, this expansion unfolds step-by-step as you diffuse each part of your original term into its new expanded format. You're ensuring that original structures are being enlarged without losing any part of the initial expression.
Knowing how to expand polynomials allows you to delve deeper into algebra, unraveling complex expressions while maintaining their integrity.

• Begins with the application of distributive property.
• Results in a more detailed expression.
• Helps decipher complex algebraic relationships.

Algebraic Multiplication

When multiplying algebraic expressions, think of it as combining different flavors to create a single cohesive dish. Each letter or base is like an ingredient, and the exponents are their quantities.
As you multiply, you add the exponents of like bases. For instance, multiplying \(2xy \cdot 5xy^2\) goes like this: You multiply the coefficients (2 and 5) to get 10, and add the exponents of x (1+1 = 2) and y (1+2 = 3), resulting in \(10x^2y^3\).
It's essential for algebraic multiplication to keep track of these terms and carefully combine them to create your full expression.

• Multiply coefficients first, then handle variables by adding exponents.
• Ensure like terms are combined to form a consistent expression.
• This fundamental skill is crucial for success in algebra and beyond.